Generic expansion and Skolemization in NSOP 1 theories

Abstract

We study expansions of NSOP 1 theories that preserve NSOP 1 . We prove that if T is a model complete NSOP 1 theory eliminating the quantifier ∃ ∞ , then the generic expansion of T by arbitrary constant, function, and relation symbols is still NSOP 1 . We give a detailed analysis of the special case of the theory of the generic L-structure, the model companion of the empty theory in an arbitrary language L. Under the same hypotheses, we show that T may be generically expanded to an NSOP 1 theory with built-in Skolem functions. In order to obtain these results, we establish strengthenings of several properties of Kim-independence in NSOP 1 theories, adding instances of algebraic independence to their conclusions.Date: September 18, 2018.

Independence in Generic Incidence Structures

“…The result of Proposition 4.25 can be easily adjusted to take place over any set of parameters, instead of over the empty set. The equivalence of forking and dividing for complete types, but not for formulas, is yet another phenomenon that T m,n shares with the Henson graphs [9] and many NSOP 1 examples from [20]. On the other hand, there is an interesting difference between the present situation and that of the Henson graphs (which are not NSOP 1 ).…”

“…The following easy fact establishes strong finite character and witnessing for | ⌣ a (since any Morley sequence in a global invariant type is | ⌣ a -independent). See also [20,Lemma 2.7].…”

“…The question of whether T m,n has a prime model seems to be a hard combinatorial problem; in the case m = n = 2, we show that it is equivalent to a longstanding open problem in the theory of projective planes (Theorem 3.8).Finally, from the point of view of classification theory, the generic theory recipe has been a fruitful source of examples of simple theories (see [6], for example). Recently, many examples of generic theories which are not simple have been shown to be NSOP 1 (see [8], [19], [20]). The theories T m,n are further examples of this phenomenon, and they provide good combinatorial examples of properly NSOP 1 theories.…”

“…Now, given k ≥ 1, let T k be the the theory of the "generic k-ary function", i.e., the model completion of the empty theory in a language consisting of a single k-ary function symbol. In [20], T k is shown to be NSOP 1 with weak elimination of imaginaries, and not simple when k ≥ 2 (arbitrary languages are considered in [20], but we focus on T k for simplicity). Like in T m,n , Kim independence in T k can be extended to arbitrary sets, and forking independence, which coincides with dividing independence, is obtained by forcing base monotonicity on Kim independence.…”

Recursive functions and existentially closed structures

“…We mention that Corollary 5.10 was independently discovered by Kruckman and Ramsey [6], who learned of the problem from an earlier unpublished version of this paper where it was posed as an open problem.…”